Given your linear programming problem, the objective is to minimize the function $Z = 25x + 15y + 410$ subject to a set of constraints. Here's the full formulation:

Objective Function:

$\text{Minimize } Z = 25x + 15y + 410$

Subject to the Constraints:

1. $x + y \leq 100$
2. $x \leq 60$
3. $y \leq 50$
4. $x + y \geq 60$

Steps to Solve:

1. Identify the feasible region: Plot the constraints on a graph with $x$ on the horizontal axis and $y$ on the vertical axis. The feasible region is the area where all constraints overlap.

2. Find the corner points of the feasible region: These are the points where the constraint lines intersect. The optimal solution in linear programming occurs at one of these corner points.

3. Calculate the value of $Z$ at each corner point: Substitute the coordinates of each corner point into the objective function to find which one gives the minimum value.

4. Determine the minimum value: The corner point with the lowest value of $Z$ is the optimal solution.

Let's solve it:

1. Constraints:

• $x + y \leq 100$
• $x \leq 60$
• $y \leq 50$
• $x + y \geq 60$

2. Find intersection points:

• Intersection of $x + y = 100$ and $x = 60$ gives $(60, 40)$.
• Intersection of $x + y = 100$ and $y = 50$ gives $(50, 50)$.
• Intersection of $x + y = 60$ and $x = 60$ gives $(60, 0)$.
• Intersection of $x + y = 60$ and $y = 50$ gives $(10, 50)$.

3. Evaluate $Z$ at each point:

1. At $(60, 40)$: $Z = 25(60) + 15(40) + 410 = 1500 + 600 + 410 = 2510$
2. At $(50, 50)$: $Z = 25(50) + 15(50) + 410 = 1250 + 750 + 410 = 2410$
3. At $(60, 0)$: $Z = 25(60) + 15(0) + 410 = 1500 + 0 + 410 = 1910$
4. At $(10, 50)$: $Z = 25(10) + 15(50) + 410 = 250 + 750 + 410 = 1410$

4. Minimum Value:

The minimum value of $Z$ occurs at $(10, 50)$ where $Z = 1410$.

Optimal Solution:

• Optimal point: $x = 10$, $y = 50$
• Minimum value of $Z$: 1410

Would you like more details on this solution, or do you have any questions?

Related Questions:

1. How can I graphically represent this linear programming problem?
2. What is the significance of the constant 410 in the objective function?
3. How does the feasible region affect the optimal solution?
4. Can we solve this problem using the Simplex method instead of graphically?
5. What if one of the constraints were non-linear? How would that change the approach?

Tip:

When solving linear programming problems graphically, always check the boundaries of your feasible region carefully, as they determine where the optimal solution lies.